Representing systems of exponential functions in polycylindrical domains

Le Khaǐ Khoǐ ; Yu. F. Korobeinik

Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 439-456

Résumé

The main result in this article is the following. Theorem. {\it Let $D_p$ $(1\leqslant p\leqslant m)$ be a bounded convex domain in the $z_p$-plane with support function $h_p(-\varphi),$ and let $\Lambda_p\overset{\mathrm{df}}=\{\lambda_k^{(p)}\}_{k=1}^\infty$ be zeros $($not necessarily simple$)$ of an exponential function $\mathscr L_p(\lambda)$ with indicator $h_p(\varphi)$ $($the function $\mathscr L_p(\lambda)$ may also have other zeros besides $\{\lambda_k^{(p)}\}_{k=1}^\infty,$ and$,$ moreover$,$ of arbitrary multiplicity$).$ Assume that $\mathscr E_{\Lambda_p}\overset{\mathrm{df}}=\{e^{\lambda_k^{(p)}z_p}\}_{k=1}^\infty$ is an absolutely representing system in $\mathscr H(D_p),$ $p=1,2,\dots,m$. Then $$ \mathscr E_{\Lambda}\overset{\mathrm{df}}=\big\{e^{\lambda_{k_1}^{(1)}z_1+\dots+\lambda_{k_m}^{(m)}z_m}\big\}_{k_1,\dots,k_m=1}^\infty $$ is an absolutely representing system in $\mathscr H(D),$ where $D=D_1\times D_2\times\dots\times D_m$ and $\mathscr H(G)$ is the space of holomorphic functions in a domain $G,$ with the topology of uniform convergence on compact subsets of $G$.} The properties of nontrivial expansions of zero in $\mathscr H(D)$ with respect to a system $\mathscr E_\Lambda$ are also studied. In particular, it is proved that if $D_p$, $\Lambda_p$, and $\mathscr L_p(\lambda)$ are the same as in the statement of the theorem, then $\mathscr E_\Lambda$ is an absolutely representing system in $\mathscr H(D)$ if and only if $\mathscr H(D)$ has a nontrivial expansion of zero with respect to the system $\mathscr E_\Lambda$. Bibliography: 9 titles.